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On the Invariance of the Tropical Zodiac 
This is an inquiry into a fundamental problem that has to do with the tropical versus sidereal zodiac controversy and how it pertains to the practice of ancient astrology. It involves the very ancient practice of granting periods of years to the signs of the zodiac according to rising times. First of all let us define what is meant by the term "rising times." As each sign rises in the east a certain number of degrees of right ascension pass over the meridian. This number of degrees constitutes the "rising times" of the sign. A period was assigned to each sign at the rate of one degree per year. For example at 40° north the tropical sign Aries rises while 18°06' pass over the meridian. Taurus rises while 21°47' pass over the meridian, etc. For the technically inclined another way of defining rising times is to compute the oblique ascension of the beginning and end of each sign and to subtract the oblique ascension or O.A. of 0 degrees of the sign from the O.A. of 30 degrees of that same sign. Oblique ascensions will be discussed below. The computation of rising times has a great antiquity. According to Neugebauer in the History of Ancient Mathematical Astronomy (HAMA) the Babylonians in an early system of computations known as "System A" created such a scheme of rising times, although the times were computed not according to modern trigonometric methods as described above but by a numerical series such as that which follows: Starting with the first 30 degree segment of the zodiac which got 20 degrees of ascension, each 30 degree segment was given exactly 4 degrees more of rising than the preceding segment up to the sixth segment. The sixth and seventh segments had the same rising times, and subsequent segments decrease at the rate of 4 degrees until the twelfth segment has the same rising time as the first. The longest rising times of the sixth and seventh 30 degree segment were each 40 degrees. We call these 30 degree segments rather than signs because according to Neugebauer these 30 degree segments began not at 0 Aries but at 10 degrees of Aries which was defined as being the location of the vernal point in the zodiacal signs. While it is tempting to assume that what we have here is measure of the vernal point in terms of a sidereal zodiac, the problem is that scholars do not agree that this vernal point was regarded as moving. The Babylonians may very well have regarded it as the permanent location of the vernal point in the zodiac. For the most part scholars do not believe that the Babylonians at this stage were aware that the vernal point precessed. According to these scholars Hipparchos the Greek discovered precession centuries after the development of System A. Other scholars such as Cyril Fagan and others who for the most part would not be regarded as "mainstream" [not necessarily to be interpreted as a criticism on my part], do believe that the Babylonians knew about precession. Pursuant to this discussion let us acknowledge one thing at this point in the discussion. If we take Neugebauer's account of these rising times at face value, then we must accept that at some level that the Babylonians who developed this system were aware of the distinction between these 30 degree "signs" and 30 degree segments measured from the vernal point. It is not quite so clear that the Babylonians were conscious "Siderealists." However, whether or not the Babylonians who developed System A were conscious "Siderealists" is not central to our discussion. Below is a table of the rising times of the 30 degree divisions from the vernal point as given in the Babylonian System A. System A R Sum Su Lg Daylight h m R1 20 20 Ar 10 12:00 R2 24 44 Ta 10 13:20 R3 28 72 Ge 10 14:08 R4 32 104 Cn 10 14:24= Max. R5 36 140 Le 10 14:08 R6 40 180 Vi 10 13:20 R7 40 220 Li 10 12:00 R8 36 256 Sc 10 10:40 R9 32 288 Sg 10 9:52 R10 28 316 Cp 10 9:36 = Min. R11 24 340 Aq 10 9:52 R12 20 360 Pi 10 10:40 The table is taken more or less directly from Neugebauer's HAMA with slight modifications in the notation. The first column simply refers to the 12 segments. The second column marked "R" contains the rising times as defined above of each of these 30 degree segments of the ecliptic. The third column gives a running total of the rising times computed from the rising of the vernal point. The fourth column gives the longitude in the zodiac in use (presumably a sidereal one) of the beginning of each 30 degree segment measured from the vernal point at 10 degrees of Aries. The fifth column gives the length of daylight that occurs when then Sun is in the particular degree. Note that when the Sun is at the vernal and autumnal points the daylight is precisely 12 hours; the daylight reaches a maximum of 14h 24m when the Sun is exactly 90 degrees later at the summer solstice point, and a minimum of 9h 36m when the Sun is at winter solstice point. Also note that the ratio of the longest daylight period to the shortest daylight period is precisely 3:2. Latitudes in the ancient world were measured according to the ratio of longest day to shortest day. Another school of Babylonians developed a second system of rising times which are associated with a "System B." In this system the 30 degree divisions are made from a vernal point which is defined as 8 degrees of Aries. This is apparently a later measurement of the vernal point in the sidereal zodiac. Below is a table of rising times. The values are different from the previous table, but the logic of the table is the same. R Sum SU LG Daylight h m R1 21 21 Ar 8 12:00 R2 24 45 Ta 8 13:12 R3 27 72 Ge 8 14:00 R4 33 105 Cn 8 14:24 = Max. R5 36 141 Le 8 14:00 R6 39 180 Vi 8 13:12 R7 39 219 Li 8 12:00 R8 36 255 Sc 8 10:48 R9 33 288 Sg 8 10:00 R10 27 315 Cp 8 9:36 = Min. R11 24 339 Aq 8 10:00 R12 21 360 Pi 8 10:48 We include this table of System B for the sake of completeness, but it will not figure in our discussion as much as System A except in one regard, the definition of the vernal point as being 8 degrees of Aries. Note that in both tables opposite 30 degree segments have rising times which total 60 degrees. Also note the following: the first and twelfth segments have the same rising times; so do the second and eleventh segments, the third and tenth segments, and so forth. This symmetry is very important for reasons that we will disclose below. First however let us look at how well the rising times of System A reflect the actual rising times of the 30 degree segments as computed according to modern methods. Rising Times of the Tropical Signs Using Modern Values Versus the Values of System A Segments R1-R12 R2-R11 R3-R10 R4-R9 R5-R8 R6-R7 System A 20 00 24 00 28 00 32 00 36 00 40 00 501 B.C.E. 20 18 23 42 29 40 34 49 36 07 35 24 The last row is computed for 501 B.C.E. using modern trigonometric methods for Babylon. The agreement is quite good for R1-R12, R2-R11, and R5-R8. The fit is cruder for the other segments. It is also worth noting that the rising times computed for these 30 degree segments measured from the vernal point (which of course correspond to the tropical signs) are quite stable over time. Below is a table showing the rising times computed using modern methods for 501 B.C.E. and 2000 C.E. for the Babylon. Rising Times of the Tropical Signs at Babylon
Ar-Pi Ta-Aq Ge-Cp Cn-Sg Le-Sc Vi-Li
501 B.C.E. 20 18 23 42 29 40 34 49 36 07 35 24
2000 C.E. 20 28 23 48 29 40 34 42 36 01 35 21
The reader will note the differences are small. For all practical purposes the rising times of the tropical signs for any particular latitude are virtually constant over time. By contrast let us look at the rising times of sidereal signs using the Fagan-Allan ayanamsha [the difference between the tropical and sidereal zodiacs] computed for the same two epochs again using modern methods. Rising Times of the Sidereal Signs at Babylon
First Half of the Zodiac
Ar Ta Ge Cn Le Vi
501 B.C.E. 21 04 25 30 31 42 35 40 35 55 35 15
Pi Aq Cp Sg Sc Li
501 B.C.E. 19 56 22 12 27 33 33 29 36 04 35 39
Second Half of the Zodiac
Ar Ta Ge Cn Le Vi
2000 C.E. 20 13 20 49 24 42 30 44 35 12 35 57
Pi Aq Cp Sg Sc Li
2000 C.E. 22 59 28 34 34 04 36 01 35 28 35 16
Upon examining the table two things become apparent. First of all the rising times of the sidereal signs are far from constant over time. Note particularly the rising time of sidereal Gemini in 501 B.C.E. which equals 31ƒ42'. But in the year 2000 C.E. Gemini's rising times will be 24ƒ 42' a difference of 7 degrees! Second the symmetry that we noted between pairs of signs R1-R12 (Ar-Pi), R2-R11 (Ta-Aq), R3-R10 (Ge-Cp), etc. is not present. In 501 B.C.E. Gemini gets 31d 42m while the sign that ought to be symmetrical with it, Capricorn, gets 27d 33m. In the year 2000 C.E. Gemini gets 24d 42m and Capricorn gets 34d 04m. This is because the symmetry of the rising times of the 30 degree segments requires that the measuring of the segments be done from one of the equinoctial points. This may require some explanation. Coordinate Table for the Beginnings of the Tropical Signs for 40 Degrees
North Latitude 270ƒ-60ƒ Tropical Longitude Long. 270 300 330 0 30 60 R.A. 270 302.18 332.09 0 27.91 57.82 Decl. -23.45 -20.16 -11.48 0 11.48 20.16 A.D. -21.34 -17.94 -9.81 0 9.81 17.94 O.A. 291.35 320.13 341.9 0 18.1 39.88 R.T. 28.78 21.77 18.1 18.1 21.77 28.78 90ƒ-240ƒ Tropical Longitude Long. 90 120 150 180 210 240 R.A. 90 122.18 152.09 180 207.91 237.82 Decl. 23.45 20.16 11.48 0 -11.48 -20.16 A.D. 21.34 17.94 9.81 0 -9.81 -17.94 O.A. 68.66 104.24 142.28 180 217.72 255.76 R.T. 35.59 38.04 37.72 37.72 38.04 35.59 In the table given above we have the following: The first row marked "Long." contains the tropical longitudes of the beginning of each sign. The second row marked "R.A." contains the right ascension of the beginning of each sign. The row marked "Decl." is declination of the ecliptic degree at the beginning of each sign. The row marked "A.D." contains the ascensional difference of the beginning of each sign. This will be explained shortly. The row marked "O.A." contains the oblique ascension of the beginning of each sign. This will also be explained shortly. And last the row marked "R.T." contains the rising times of the signs which begin at the designated longitude. First an explanation of oblique ascension. On the equator all positions on the celestial sphere, regardless of declination, rise along with their right ascensions at 0 degrees declination. This is because at the terrestrial equator the celestial equator rises in the east exactly perpendicular to the horizon, hence the term "right" ascension, "right" meaning perfectly upright. But either north or south of the terrestrial equator positions on the celestial sphere do not rise with their positions measured in right ascension. They rise along some other degree on the celestial equator. This other degree is the oblique (or slantwise) ascension of our hypothetical position on the celestial sphere. It is called oblique ascension because the celestial equator at latitudes other than 0 degrees north or south rises slantwise or obliquely in the east, the further away from 0 terrestrial latitude (the equator), the more obliquely. Therefore, the oblique ascension of position A can be defined as whatever degree on the equator may be rising when A exactly touches the horizon assuming that A is not on the celestial equator, i.e., that A has a declination not equal to 0. Ascensional difference or A.D. is a measure of the difference between the R.A. or right ascension of a point and its O.A. or oblique ascension. The formula is as follows: O.A. = R.A. - A.D. Thus the A.D. of a point is required to find the O.A. of that point. The A.D. of a point in turn is derived from the declination of the point and the terrestrial latitude of the place in question by the following formula. A.D. = arcsin(tan Decl. x tan Latitude) These relationships can be seen in the coordinate table shown above. Note from the table that arc from the O.A. of 330 degrees to the O.A. of 360 or 0 degrees is the same as the arc from the O.A. of 0 degrees to the O.A. of 30 degrees. These arcs are the rising times of tropical Pisces and Aries. This is the consequence of the following two facts: First that the arc in R.A. from 0 degrees tropical Pisces to 0 degrees tropical Aries is the same as the arc in R.A. from 0 degrees tropical Aries to 30 degrees tropical Aries (or 0 degrees Taurus). The second fact is that the declination of 0 degrees tropical Pisces is the exact opposite of the declination of 30 degrees tropical Aries, the first being -11.477 degrees, the second being +11.477. This in turn causes the A.D. of 0 degrees tropical Pisces to be the exact opposite of the A.D. of 30 degrees tropical Aries. Only if two points are symmetrical with respect to the equinoxes can they possess this symmetry of arcs in O.A. which in turn produces the symmetrical rising times of signs which are equidistant from the equinoxes. This symmetry can occur in a sidereal zodiac only when the vernal point is at exactly 0 degrees of a sign. The Babylonians of Systems A and B knew this which is why they measured the rising times of 30 degree arcs from the vernal point rather than from 0 degrees of Aries in a zodiac in which the vernal point was not at 0 Aries (or any other sign). This tells us something very important which seems to have escaped the notice of nearly everyone. The Babylonians of Systems A and B had at least two twelvefold divisions of the ecliptic into 30 degree divisions: One was made from a point which was 10 or 8 degrees prior to the vernal point. This "zodiac" may or may not have been consciously sidereal. The second was a "zodiac" which was measured from the vernal point and which clearly was consciously tropical. However, was this "tropical zodiac" actually used for any astrological purpose? For that matter was the other, possibly consciously sidereal, zodiac used for astrological purposes? The usual answer to the latter question is yes, but this is a matter which we will need to examine further. Likewise the answer to the first question is usually said to be no. But the fact remains that there was a twelvefold equal tropical division of the ecliptic in Babylonian times along with the probably sidereal one. This paves the way for the next stage of things. The central difficulty that has not been dealt with in this controversy is this question. What was the practice of astrologers when they came to do astrology more or less as we know it? What zodiac did they use, and were they really conscious of what they were doing? For it is not enough to show that early charts were computed using a sidereal zodiac if the people who cast them were not aware that they were using a sidereal zodiac. And the entire controversy becomes moot if it can be shown that for all practical purposes the two zodiacs were not distinguished. The oldest known birthchart has been dated by Sachs to 410 B.C.E. It is a cuneiform chart with no degrees given, only sign positions, also no Ascending degree. Computations using both tropical and sidereal zodiacs give the same signs for all of the planets so listed. This chart therefore is of no use in determining the zodiac in use. The next several charts in cuneiform date from the 3rd century B.C.E. These do contain degrees for individual planets and these positions are reasonably consistent with positions in the Fagan-Allen sidereal zodiac. However, they could also conceivably be computed in a the zodiac in which the vernal point is fixed at 8 Aries, the System B zodiac. None of these charts have Ascendants which means that if these charts are accurate exemplars of the chart technology of the age, we are dealing with a very primitive form of horoscopy, not the sophisticated one that appears in later Greek astrology. Yet we are well within the period that appears to be the date for the Nechepso-Petosiris text which already shows an advanced horoscopic technique. Could Egyptian horoscopy have already outstripped the Babylonian art on which it was undoubtedly based? The following table shows the close coinciding of the zodiacs through the period in which horoscopic astrology comes into being and begins truly to flower. The longitudes are the positions of the vernal point given in terms of the sidereal zodiac of Fagan-Allen. It is clear that only the most precise astronomical computations could allow us to clearly distinguish the two zodiacs in the main period of Greek astrology. 301 B.C.E. 7 Ar 13 201 B.C.E. 5 Ar 50 101 B.C.E. 4 Ar 27 1 B.C.E. 3 Ar 04 100 C.E. 1 Ar 41 200 C.E. 0 Ar 18 300 C.E. 28 Pi 55 400 C.E. 27 Pi 32 So do we see evidence of the astrologers of this period having a clear idea of their zodiac? Of course there is the example of Ptolemy (roughly 175 C.E.) who explicitly states that the zodiac begins with the vernal point. And we also have his witness that Hipparchos supported the same position about 300 years before. But was this issue clearly delineated in the minds of astrologers in general? That is the question. Let's explore the answer to this question first of all by means of the ascensional times which we have been describing here. When astrology clearly emerges into the documentable daylight, the early centuries C.E. (A.D.), the Babylonian doctrine of ascensional times comes along with it. Vettius Valens was a younger contemporary of Ptolemy, about 175 C.E., but his astrological style is clearly derived from a different tradition than Ptolemy's, one at least as old. In the Anthology, Book I, chapter 6, there is the following passage.
These are the standard System A ascensional times. However, as we know from Books VIII and IX, even though Valens used System A ascensional times, he set the vernal point at 8 degrees of Aries, the position used in System B. And in addition he and the others who used this system did not make the distinction that the Babylonians seemed to have made. He did not locate the 30 degree tropical divisions which gave rise to the ascensional times at 8 degrees of the signs but at 0 degrees of the signs thus identifying the signs of the zodiac with the divisions that gave rise to the ascensional times. And there is no evidence that this was an innovation of Valens, but rather a common convention among the astrologers of the period. This clearly indicates that these astrologers did not have a clear idea of the distinction between sidereal and tropical zodiacs, or that they did not consider the distinction to be important, not a remarkable position given the fact the two zodiacs did nearly coincide at the time. Also characteristic of the texts of the time are references to the signs in terms that we would clearly recognize as based on seasonal criteria along with factors that we clearly recognize as sidereal. Consider the following description from Valens Book I. (Robert Schmidt Trans.)
Star Lg. Ptol. Lg. Anon. Sid. Lg. Aldebaran 13 Ta 15 Ta 15 Ta Pollux 27 Ge 29 Ge 29 Ge Regulus 3 Le 20 Le 5 Le Spica 27 Vi 29 Vi 29 Vi Antares 13 Sc 15 Sc 15 Sc The difference between the Ptolemaic positions and those of the Anonymous is exactly the 2 degrees that one would get using the erroneous value for precession of Ptolemy's. Only Regulus is different in the Anonymous but the corrected Ptolemy position gives exactly the sidereal position. The coinciding of the positions given by the Anonymous with the sidereal positions is a coincidence! And other authors that are sometimes cited as siderealists are from the same general period, Hephaistio of Thebes, and Firmicus Maternus. On page 10 of The Zodiac: A Historical Survey by Robert Powell the author cites a passage from Neugebauer's HAMA as evidence for the Anonymous being a siderealist. Unfortunately the passage in question is one in which Neugebauer is dating this author and another author named Cleomedes to the 4th century by showing that their values for star positions are derived from correcting Ptolemy's positions using his precessional constant! One wonders how much of the evidence for the sidereal zodiac among the Greeks comes from similarly questionable research. When did the confusion of the zodiacs clearly end? There is no simple answer to this question. However, it is clear that the kind of confusion that we are documenting here survived among the Hindus. The following is from Varahamahira's Brihat Jataka, chapter 1, sloka 19. "The measures of the first six signs are represented by the numbers 20, 24, 28, 32, 36, and 40 respectively. The same figures taken in inverse order give the measures of the second six signs." These are our old friends the System A rising times for Babylon again; and again, just as in Valens, they are identified with the signs of the zodiac, not a separate set of 30 degree divisions having no fixed relation to the signs of the zodiac. And again they are symmetrical with respect to 0 degree Aries, something that can only happen in a tropical zodiac. Was this eminent figure of the Hindu tradition a tropicalist? Apparently so. In another early Hindu work, the Yavana Jataka, we also find symmetrical rising times, indicating a tropical zodiac although these rising times at least are recomputed for India. We now have to deal with a fundamental question: which zodiac would have seemed the more reasonable to the ancients? This is not a trivial question because it has been argued by moderns that a sidereal zodiac would seem on the face of it to be more rational. After all while the stars do have a small motion relative to each other, their proper motions, most of their apparent motion is in fact due to the backwards motion of the vernal point with respect to the fixed stars. It is obviously more rational from a modern point of view to consider the one point as being in motion rather than the many points. Also modern astronomy regards the solar system from the point of view of the Sun rather than the Earth, and it is therefore more reasonable to regard the stars as more or less stationary than it is to so regard one single Earth-related point, the vernal point. But these are the criteria of moderns. Would they have been the criteria of the ancients? Certainly it would have been easier for them to measure positions with regard to fixed stars and we have abundant evidence of the practice. But we also know that various persons among the ancients were in fact quite capable of locating the cardinal tropical points as is evidenced by the number of allignments to the rising positions of the these points all over Europe. The problem is that most of the ancients regarded the Earth as being completely stationary. There were exceptions such as some Pythagoreans, Aristarchos (who actually posited a heliocentric theory), and at least one Hindu, Aryabhata. The most common view was that there were eight spheres surrounding the earth. The eighth sphere held the fixed stars and also rotated about the stationary earth once every twenty-four hours, what was later called the Primum Mobile. The other seven are the spheres of the seven planets. Here is the description of the creation of these spheres from the Timaeus of Plato. "And thus the whole mixture out of which he cut these portions was all exhausted by him. This entire compound he divided lengthways into two parts, which he joined to one another at the centre like the letter X, and bent them into a circular form, connecting them with themselves and each other at the point opposite to their original meeting-point; and, comprehending them in a uniform revolution upon the same axis, he made the one the outer and the other the inner circle. Now the motion of the outer circle he called the motion of the same, and the motion of the inner circle the motion of the other or diverse. The motion of the same he carried round by the side to the right, and the motion of the diverse diagonally to the left. And he gave dominion to the motion of the same and like, for that he left single and undivided; but the inner motion he divided in six places and made seven unequal circles having their intervals in ratios of two-and three, three of each, and bade the orbits proceed in a direction opposite to one another; and three [Sun, Mercury, Venus] he made to move with equal swiftness, and the remaining four [Moon, Saturn, Mars, Jupiter] to move with unequal swiftness to the three and to one another, but in due proportion. The equatorial motion of the eighth sphere is designated the circle of the same or invariant, while the other seven circles, those of the planets, are derived from the circle "of the other or diverse." Then, when precession became a clearly understood doctrine, to the eighth sphere was added a ninth sphere which became the primum mobile, and the old eighth sphere held the fixed stars. And these were perceived as moving with respect to the sphere of the primum mobile. This fact clearly demonstrates that the fixed stars were conceived as being in motion with respect to the primum mobile; the components of the primum mobile are the vernal point, the celestial equator, the other equinox, and the solstices. Nor is this all. We have already shown that the rising times in the tropical zodiac, a critical feature of the ancient system, is nearly invariant over time while the rising times of the constellations are not. And those who measured the positions of the Sun at dawn along the horizon would have noticed that the Sun always rose at the same position along the horizon at the same time of year and that the maximum northerly and southerly positions along the horizon were virtually invariant. And even in Babylonia we know that the first constellation of the stellar zodiac was the one which rose at dawn in the spring. Clearly even they, insofar as they knew that the stars and vernal point were moving with respect to each other, would have regarded the stars as being in motion, not the vernal point. From what we have seen it is clear that the Babylonians had two divisions of thirty degrees, one corresponding to the constellations which they may or may not have known were moving with respect to the vernal point, and another which was fixed with respect to the vernal point with the vernal point at 0 degrees. The Greeks did not invent the tropical zodiac as often charged. All they did was to give the names of the constellations to the tropical signs. We actually have no way of knowing at this point what the ancients regarded as the astrologically effective set of divisions, or even if they did regard only one set as being effective. Later generations right up to modern times in the West regarded both sets as being effective, but the later medieval and renaissance astrologers did not regard the constellations as being equal. They regarded them as asterisms of unequal extent. To conclude: I do not assert that the ancients were tropicalists, nor do I assert that they were siderealists. I assert that whatever they may have known about precession they tended not to make the distinction, and when they did, they would have been just as likely to give precedence to the tropical as the sidereal for divinatory purposes. After all the pictorial constellations were only physical plane images which roughly corresponded to the ideal, mathematical reality which would have been represented by the tropical system. But fundamentally I believe we have to regard the tropical-sidereal controversy as yet another example of a historical pseudo-problem created by anachronistically projecting a modern problem with modern points of view back onto the ancients. It was not a problem with which the ancients were seriously concerned. Given the limits of their computational accuracy, both systems would have given them the same results. This is a question that we have to solve for ourselves. An appeal to history will not work. |
